By Fred Diamond, Payman L. Kassaei, Minhyong Kim

Automorphic types and Galois representations have performed a relevant function within the improvement of recent quantity conception, with the previous coming to prominence through the distinguished Langlands application and Wiles' facts of Fermat's final Theorem. This two-volume assortment arose from the 94th LMS-EPSRC Durham Symposium on 'Automorphic types and Galois Representations' in July 2011, the purpose of which used to be to discover contemporary advancements during this sector. The expository articles and learn papers around the volumes replicate fresh curiosity in p-adic tools in quantity idea and illustration thought, in addition to contemporary development on issues from anabelian geometry to p-adic Hodge idea and the Langlands software. the themes coated in quantity comprise curves and vector bundles in p-adic Hodge concept, associators, Shimura forms, the birational part conjecture, and different subject matters of up to date curiosity.

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**Example text**

In fact, if x ∈ Bb,+ and r ≥ r > 0 then vr (x) ≥ r vr (x). 3) Thus, if 0 < ρ ≤ ρ < 1 we have + + B+ [ρ,ρ ] = Bρ ⊂ Bρ b,+ where for a compact interval I ⊂]0, 1[ we note B+ I for the completion of B + + with respect to the (| · |ρ )ρ∈I , and Bρ := B{ρ} . One deduces that for any ρ0 ∈]0, 1[, B+ ρ0 is stable under ϕ and B+ = ϕ n B+ ρ0 n≥0 the biggest sub-algebra of B+ ρ0 on which ϕ is bijective. Suppose E = Q p and choose ρ ∈ |F × |∩]0, 1[. Let a ∈ F such that |a| = ρ. D. hull of the ideal W (O F )[a] of W (O F ) ⊗Z p Q p .

18. When E = Fq ((π )), replacing WO E (O F ) by O F z in the preceding definitions (we set z = π ) there is an identification |Y | = |D∗ |. In fact, according to Weierstrass, any irreducible primitive f ∈ O F z has a unique irreducible unitary polynomial P ∈ O F [z] in its O F z × -orbit satisfying: P(0) = 0 and the roots of P have absolute value < 1. Then for y ∈ |D∗ |, deg(y) = [k(y) : F] and y is the distance from y to the origin of the disk D. 2. Background on the ring R For an O E -algebra A set R(A) = x (n) n≥0 | x (n) ∈ A, x (n+1) q = x (n) .

3. 4. For ∈ m F \ {0} and u = [ [ ]Q one 1/q ] Q has ϕ n (u ) = [π n ]LT ([ ] Q ) [π n−1 ]LT ([ ] Q ) and thus + (u ) = n≥0 = = ϕ n (u ) π 1 π[ 1/q ] π[ 1/q ] Q 1 Q . lim π −n [π n ]LT [ ] Q n→+∞ logLT [ ] Q ) where logLT is the logarithm of the Lubin–Tate group law LT . Moreover, one can take − (u ) = π[ 1/q ]Q and thus (u ) = logLT ([ ] Q ). 3). In fact we have the following period isomorphism. 61. The logarithm induces an isomorphism of E-Banach spaces mF , + LT ∼ −→ Bϕ=π −→ logLT [ ] Q . 62.